You are right a fox ain't very big and yes a .22lr can do a fine job at short ranges if you do your part. Many hunters feel the .22M is ideal for fox out to 100 yards or maybe a bit more. A .22lr shooting CCI Velocitors is about the same at 50 yards. Don't go crossing the .44 off the list. Use a cast bullet load (if using factory ammo think cowboy loads) that has as flat of a point as possible. You'll get an exit every time but shouldn't do any more damage than other centerfires and less than many.
Cept for proper centerfires, If ya got a shotgun that's your best bet. 12ga, full choke or tighter, #4 buck will do the job. Fox ain't known to just stand around and give great shot opportunities and a quick pointing shotgun works great.